Note: the answer isn't Jar Jar. You may hate the character, but plotwise, he's mostly harmless. Cutting him may reduce the irritation, but it doesn't improve the story.
The pod race is the lowest hanging fruit in the forest. This vast, pointless exercise tells us nothing about the characters, provides no meaningful tension, and gives us nothing to fear. It's no more than a purposeful roller-coaster ride.
What does the podrace lack? It lacks ANGER and it lacks ANGER leading to THE FORCE.
I'll rewrite. Little Anakin starts his podrace. When his pod goes out of control through sabotage, the connector cable snapping about, he uses THE FORCE to grab the loose cable. Ah, we say, this kid does have it. He does have the force, and he's already strong enough to grab things. This is his aha moment. He realize that ANGER connects him to the force. With the ANGER, he uses THE FORCE to make his podracer go faster. He uses ANGER/THE FORCE to catch up. He uses ANGER/THE FORCE to choke the champion podracer, causing his wreck.
This is an improvement because the audience sees that Anakin really is special, and later on, when Yoda forbids Anakins training, the audience udnerstands why this as wise. To remove the scene would now be to gut Anakins awakening, and our fundamental understanding of how this character is going astray.
So where else can we use the temper? Mom. We can have mom afraid of Anakin. She doesn't even need to say anything different, she just needs to act like the child is a fearful menace. His temper has gotten her sold multiple times. Only Sebulba, so immune to the force, seems to be able to handle the kid, and even he's getting tired of the brat.
Another solution would be to advance the age of Anakin by some years. That would take no changes to the script and get us a slightly older actor, and therefore, a stronger performance.
We can also change how Anakin appears in the camera. By making lighting and camera angle choices, we can emphasize an intensity to the kid. Importantly, we can make him seem both great and sinister at the same time. In fact, we need to see his hero capacity shining out just as much as we see his sinister side screaming out. He'll be a hero who's always walking the line, striving for the light side, but always tempted by the dark.
An important principle is that what happens here must echo into the main film and give those scenes more meaning, more gravitas. Those scenes with the force must bind all the films together.
The other way to improve the film is to actually have Qui Gon's character matter. He seems like he matters, but none of his ideas or observations actually come out in the film, nor do they influence any of the following films. In sad truth, he could have been a robot with a light saber spouting out prerecorded lines. That doesn't make a good trilogy. He needs a better use. His touch must be present through all the prequals and into the classic films. The victory of the light side must come from his observations. He is the one who talks about the all-connected force that connects us all. He is the one who first throws down his light saber and says, "I will not fight you. If you strike me down, I will become more powerful than you can possibly imagine."
Yoda needs to fit into this narrative. We must change Yoda a bit. In the beginning, Yoda only talks about the force as a tool for the Jedi to use, as if he is a master craftsman, and only later adopts the idea of the all flowing force of Qui Gon. This history explains why, in Empire, Yoda is so reluctant to be a teacher, not only because Luke is too old, but under Yoda's teaching the Jedi failed. Yoda failed as a teacher and failed as a learner. He now doubts that he can succeed this time with a student that should be just as terrible as Vader. His reluctance shows his own doubts.
At first, Ben retreats to Yoda's teaching, teaching Anakin traditionally, but by the end of the second film, will come around to Qui Gon's teachings. From Ben's point of view, the first trilogy reveals how Master Qui Gon was right, and it's this teaching which allows him to surpass Anakin despite Anakin's epic anger. He passes this teaching to Luke, and so does Yoda. In Empire, Luke fails to learn his lesson at the dark tree and draws his light saber. Only at the end of Return of the Jedi does Luke throw down his weapon, trusting himself entirely to the living force as Qui Gon did and as Ben did. Vader has always though that Luke learned the old style of Jedi training from Ben, but with Luke's action, he sees Qui Gon, and he understands at that moment that Luke is part of the living force tradition. Qui Gon, for a brief moment, was like his own father. Qui Gon gave everything so that Anakin could live. At that moment, Vader has the same decision to make about his own son. Who's side would he choose? He chooses the living force.
With those elements in there, we'd have a far stronger story, one that arcs over the prequals and clear into the main three films. These events would give meaning to the central films without taking away the emotions or gravitas of any of the existing scenes.